Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram

Finite size effects on locating conformational transitions for macromolecules

It has been shown from simulation and experiment that locations of peaks in structural and thermodynamic quantities accompanying "phase" transitions of a single macromolecule (collapse or crystallization/melting) do not coincide. Thus, for chains with finite lengths these different measures yield apparently different results for transition temperatures. To resolve this issue we use scaling, verified by computer simulations, to conclusively show that these different locations for peak positions are simply a consequence of the finite chain length, as has been conjectured previously.

A multiscale approach to sampling nascent peptide chains in the ribosomal exit tunnel

We present a new multiscale method that combines all-atom molecular dynamics with coarse-grained sampling, towards the aim of bridging two levels of physiology: the atomic scale of protein side chains and small molecules, and the huge scale of macromolecular complexes like the ribosome. Our approach uses all-atom simulations of peptide (or other ligand) fragments to calculate local 3D spatial potentials of mean force (PMF). The individual fragment PMFs are then used as a potential for a coarse-grained chain representation of the entire molecule. Conformational space and sequence space are sampled efficiently using generalized ensemble Monte Carlo. Here, we apply this method to the study of nascent polypeptides inside the cavity of the ribosome exit tunnel. We show how the method can be used to explore the accessible conformational and sequence space of nascent polypeptide chains near the ribosome peptidyl transfer center (PTC), with the eventual aim of understanding the basis of specificity for co-translational regulation. The method has many potential applications to predicting binding specificity and design, and is sufficiently general to allow even greater separation of scales in future work.

Strong violation of critical phenomena universality: Wang-Landau study of the two-dimensional Blume-Capel model under bond randomness

We study the pure and random-bond versions of the square lattice ferromagnetic Blume-Capel model, in both the first-order and second-order phase transition regimes of the pure model. Phase transition temperatures, thermal and magnetic critical exponents are determined for lattice sizes in the range L=20-100 via a sophisticated two-stage numerical strategy of entropic sampling in dominant energy subspaces, using mainly the Wang-Landau algorithm. The second-order phase transition, emerging under random bonds from the second-order regime of the pure model, has the same values of critical exponents as the two-dimensional Ising universality class, with the effect of the bond disorder on the specific heat being well described by double-logarithmic corrections, our findings thus supporting the marginal irrelevance of quenched bond randomness. On the other hand, the second-order transition, emerging under bond randomness from the first-order regime of the pure model, has a distinctive universality class with nu=1.30(6) and beta/nu = 0.128(5) . These results amount to a strong violation of universality principle of critical phenomena, since these two second-order transitions, with different sets of critical exponents, are between the same ferromagnetic and paramagnetic phases. Furthermore, the latter of these two sets of results supports an extensive but weak universality, since it has the same magnetic critical exponent (but a different thermal critical exponent) as a wide variety of two-dimensional systems with and without quenched disorder. In the conversion by bond randomness of the first-order transition of the pure system to second order, we detect, by introducing and evaluating connectivity spin densities, a microsegregation that also explains the increase we find in the phase transition temperature under bond randomness.

Improving Wang-Landau sampling with adaptive windows

Wang-Landau sampling (WLS) of large systems requires dividing the energy range into "windows" and joining the results of simulations in each window. The resulting density of states (and associated thermodynamic functions) is shown to suffer from boundary effects in simulations of lattice polymers and the five-state Potts model. Here, we implement WLS using adaptive windows. Instead of defining fixed energy windows (or windows in the energy-magnetization plane for the Potts model), the boundary positions depend on the set of energy values on which the histogram is flat at a given stage of the simulation. Shifting the windows each time the modification factor f is reduced, we eliminate border effects that arise in simulations using fixed windows. Adaptive windows extend significantly the range of system sizes that may be studied reliably using WLS.

Optimal modification factor and convergence of the Wang-Landau algorithm

We propose a strategy to achieve the fastest convergence in the Wang-Landau algorithm with varying modification factors. With this strategy, the convergence of a simulation is at least as good as the conventional Monte Carlo algorithm, i.e., the statistical error vanishes as 1/sqrt t, where t is a normalized time of the simulation. However, we also prove that the error cannot vanish faster than 1/t . Our findings are consistent with the 1/t Wang-Landau algorithm discovered recently, and we argue that one needs external information in the simulation to beat the conventional Monte Carlo algorithm.

First-order phase transitions: a study through the parallel tempering method

We study the applicability of the parallel tempering (PT) method in the investigation of first-order phase transitions. In this method, replicas of the same system are simulated simultaneously at different temperatures, and the configurations of two randomly chosen replicas can occasionally be interchanged. We apply the PT method for the Blume-Emery-Griffiths model, which displays strong first-order transitions at low temperatures. A precise estimate of coexistence lines is obtained, revealing that the PT method may be a successful tool for the characterization of discontinuous transitions.

Structure and stability of isotropic states of hard platelet fluids

We study the thermodynamics and the pair structure of hard, infinitely thin, circular platelets in the isotropic phase. Monte Carlo simulation results indicate a rich spatial structure of the spherical expansion components of the direct correlation function, including nonmonotonical variation of some of the components with density. Integral equation theory is shown to reproduce the main features observed in simulations. The hypernetted chain closure, as well as its extended versions that include the bridge function up to second and third order in density, perform better than both the Percus-Yevick closure and Verlet bridge function approximation. Using a recent fundamental measure density functional theory, an analytic expression for the direct correlation function is obtained as the sum of the Mayer bond and a term proportional to the density and the intersection length of two platelets. This is shown to give a reasonable estimate of the structure found in simulations, but to fail to capture the nonmonotonic variation with density. We also carry out a density functional stability analysis of the isotropic phase with respect to nematic ordering and show that the limiting density is consistent with that where the Kerr coefficient vanishes. As a reference system, we compare to simulation results for hard oblate spheroids with small, but nonzero elongations, demonstrating that the case of vanishingly thin platelets is approached smoothly.

Macroscopically ordered water in nanopores

Water confined into the interior channels of narrow carbon nanotubes or transmembrane proteins forms collectively oriented molecular wires held together by tight hydrogen bonds. Here, we explore the thermodynamic stability and dipolar orientation of such 1D water chains from nanoscopic to macroscopic dimensions. We show that a dipole lattice model accurately recovers key properties of 1D confined water when compared to atomically detailed simulations. In a major reduction in computational complexity, we represent the dipole model in terms of effective Coulombic charges, which allows us to study pores of macroscopic lengths in equilibrium with a water bath (or vapor). We find that at ambient conditions, the water chains filling the tube are essentially continuous up to macroscopic dimensions. At reduced water vapor pressure, we observe a 1D Ising-like filling/emptying transition without a true phase transition in the thermodynamic limit. In the filled state, the chains of water molecules in the tube remain dipole-ordered up to macroscopic lengths of approximately 0.1 mm, and the dipolar order is estimated to persist for times up to approximately 0.1 s. The observed dipolar order in continuous water chains is a precondition for the use of nanoconfined 1D water as mediator of fast long-range proton transport, e.g., in fuel cells. For water-filled nanotube bundles and membranes, we expect anti-ferroelectric behavior, resulting in a rich phase diagram similar to that of a 2D Coulomb gas.

Phase transition in Heisenberg stacked triangular antiferromagnets: end of a controversy

By using the Wang-Landau flat-histogram Monte Carlo (MC) method for very large lattice sizes never simulated before, we show that the phase transition in the frustrated Heisenberg stacked triangular antiferromagnet is of first order, contrary to results of earlier MC simulations using old-fashioned methods. Our result lends support to the conclusion of a nonperturbative renormalization group performed on an effective Hamiltonian. It puts an end to a 20-year -long controversial issue.

Elasticity of a polydisperse hard-sphere crystal

A general Monte Carlo simulation method of calculating the elastic constants of a polydisperse hard-sphere colloidal crystal is developed. Elastic constants of a size-polydisperse hard-sphere fcc crystal are calculated. The pressure and three elastic constants ( C11, C12, and C44 ) increase significantly with the polydispersity. It was also found from extrapolation that there is a mechanical terminal polydispersity above which a fcc crystal will be mechanically unstable.

Effect of polydispersity on the relative stability of hard-sphere crystals

By extending the nonequilibrium potential refinement algorithm and lattice switch method to the semigrand ensemble, the semigrand potentials of the fcc and hcp structures of polydisperse hard-sphere crystals are calculated with the bias sampling scheme. The result shows that the fcc structure is more stable than the hcp structure for polydisperse hard-sphere crystals below the terminal polydispersity.

Adaptive biasing force method for scalar and vector free energy calculations

In free energy calculations based on thermodynamic integration, it is necessary to compute the derivatives of the free energy as a function of one (scalar case) or several (vector case) order parameters. We derive in a compact way a general formulation for evaluating these derivatives as the average of a mean force acting on the order parameters, which involves first derivatives with respect to both Cartesian coordinates and time. This is in contrast with the previously derived formulas, which require first and second derivatives of the order parameter with respect to Cartesian coordinates. As illustrated in a concrete example, the main advantage of this new formulation is the simplicity of its use, especially for complicated order parameters. It is also straightforward to implement in a molecular dynamics code, as can be seen from the pseudocode given at the end. We further discuss how the approach based on time derivatives can be combined with the adaptive biasing force method, an enhanced sampling technique that rapidly yields uniform sampling of the order parameters, and by doing so greatly improves the efficiency of free energy calculations. Using the backbone dihedral angles Phi and Psi in N-acetylalanyl-N'-methylamide as a numerical example, we present a technique to reconstruct the free energy from its derivatives, a calculation that presents some difficulties in the vector case because of the statistical errors affecting the derivatives.

An entropic perspective of protein stability on surfaces

The interaction of proteins with surfaces regulates numerous processes in nature, science, and technology. In many applications, it is desirable to place proteins on surfaces in an active state, and tethering represents one manner in which to accomplish this. However, a clear understanding of how tether placement and design affects protein activity is lacking. Available theoretical models predict that proteins will be stabilized when tethered to substrates. Such models suggest that the surface reduces the number of states accessible to the unfolded state of the protein, thereby reducing the entropic cost of folding on the surface compared to the bulk case. Recent studies, however, have shown that this stabilization is not always seen. The purpose of this article is to determine the validity of the theory with a thorough thermodynamic analysis of the folding of peptides attached to surfaces. Configuration-temperature-density-of-states Monte Carlo simulations are used to examine the behavior of four different peptides of different secondary and tertiary structure. It is found that the surface does reduce the entropic cost of folding for tethered peptides, as the theory suggests. This effect, however, does not always translate into improved stability because the surface may also have a destabilizing enthalpic effect. The theory neglects this effect and assumes that the enthalpy of folding is the same on and off the surface. Both the enthalpic and entropic contributions to the stability are found to be topology- and tether-placement-specific; we show that stability cannot be predicted a priori. A detailed analysis of the folding of protein A shows how the same protein can be both stabilized and destabilized on a surface depending upon how the tethering enhances or hinders the ability of the peptide to form correct tertiary structures.

Determination of free energy profiles by repository based adaptive umbrella sampling: bridging nonequilibrium and quasiequilibrium simulations

We propose a new adaptive sampling approach to determine free energy profiles with molecular dynamics simulations, which is called as "repository based adaptive umbrella sampling" (RBAUS). Its main idea is that a sampling repository is continuously updated based on the latest simulation data, and the accumulated knowledge and sampling history are then employed to determine whether and how to update the biasing umbrella potential for subsequent simulations. In comparison with other adaptive methods, a unique and attractive feature of the RBAUS approach is that the frequency for updating the biasing potential depends on the sampling history and is adaptively determined on the fly, which makes it possible to smoothly bridge nonequilibrium and quasiequilibrium simulations. The RBAUS method is first tested by simulations on two simple systems: a double well model system with a variety of barriers and the dissociation of a NaCl molecule in water. Its efficiency and applicability are further illustrated in ab initio quantum mechanics/molecular mechanics molecular dynamics simulations of a methyl-transfer reaction in aqueous solution.

Structure of molecular liquids: hard rod-disk mixtures

The structure of hard rod-disk mixtures is studied using Monte Carlo simulations and integral equation theory, for a range of densities in the isotropic phase. By extension of methods used in single component fluids, the pair correlation functions of the molecules are calculated and comparisons between simulation and integral equation theory, using a number of different closure relations, are made. Comparison is also made for thermodynamic data and phase behavior.

Phase behavior of a family of continuous two-dimensional n -vector models with n=2, 3, and 4

In this work we investigate the phase behavior of a family of continuous bidimensional n -vector models (with n=2, 3, and 4) using Monte Carlo simulation. In all cases we detect the presence of a defect mediated order-disorder transition of the Berezinskii-Kosterlitz-Thouless type with critical temperatures that decrease with the spin dimensionality. Coupled with the order-disorder transition a gas-liquid equilibrium is found at low temperatures. Here one observes that the stability region of the liquid phase shrinks with the growing spin dimensionality, in parallel with a decrease in magnitude of the angular averaged spin-spin interaction.

Efficient parallel tempering for first-order phase transitions

We present a Monte Carlo algorithm that facilitates efficient parallel tempering simulations of the density of states g(E) . We show that the algorithm eliminates the supercritical slowing down in the case of the Q=20 and Q=256 Potts models in two dimensions, typical examples for systems with extreme first-order phase transitions. As recently predicted, and shown here, the microcanonical heat capacity along the calorimetric curve has negative values for finite systems.

Enhanced sampling via strong coupling to a heat bath: relationship between Tsallis and multicanonical algorithms

We show that Tsallis and multicanonical statistical mechanics are equivalent under specific conditions and that they describe a system strongly coupled to a heat bath. The concept of the strong coupling to a heat bath, in which energy fluctuation is larger than that in the canonical ensemble [J. Chem. Phys. 119, 7075 (2003)], plays a key role in relating Tsallis formalism to multicanonical formalism. The equivalence between these formalisms allows us to obtain an appropriate q parameter in the Tsallis algorithm to enhance the sampling in the phase space in a manner similar to the multicanonical algorithm. An enhanced sampling in the configurational space by use of the strong coupling formalism is demonstrated in a Lennard-Jones fluid.

Potential energy and free energy landscapes

Familiar concepts for small molecules may require reinterpretation for larger systems. For example, rearrangements between geometrical isomers are usually considered in terms of transitions between the corresponding local minima on the underlying potential energy surface, V. However, transitions between bulk phases such as solid and liquid, or between the denatured and native states of a protein, are normally addressed in terms of free energy minima. To reestablish a connection with the potential energy surface we must think in terms of representative samples of local minima of V, from which a free energy surface is projected by averaging over most of the coordinates. The present contribution outlines how this connection can be developed into a tool for quantitative calculations. In particular, stepping between the local minima of V provides powerful methods for locating the global potential energy minimum, and for calculating global thermodynamic properties. When the transition states that link local minima are also sampled we can exploit statistical rate theory to obtain insight into global dynamics and rare events. Visualizing the potential energy landscape helps to explain how the network of local minima and transition states determines properties such as heat capacity features, which signify transitions between free energy minima. The organization of the landscape also reveals how certain systems can reliably locate particular structures on the experimental time scale from among an exponentially large number of local minima. Such directed searches not only enable proteins to overcome Levinthal's paradox but may also underlie the formation of "magic numbers" in molecular beams, the self-assembly of macromolecular structures, and crystallization.

Simulation via direct computation of partition functions

In this paper, we demonstrate the efficiency of simulations via direct computation of the partition function under various macroscopic conditions, such as different temperatures or volumes. The method can compute partition functions by flattening histograms, through, for example, the Wang-Landau recursive scheme, outside the energy space. This method offers a more general and flexible framework for handling various types of ensembles, especially ones in which computation of the density of states is not convenient. It can be easily scaled to large systems, and it is flexible in incorporating Monte Carlo cluster algorithms or molecular dynamics. High efficiency is shown in simulating large Ising models, in finding ground states of simple protein models, and in studying the liquid-vapor phase transition of a simple fluid. The method is very simple to implement and we expect it to be efficient in studying complex systems with rugged energy landscapes, e.g., biological macromolecules.